'Might I have a quiet word in your ear? I wondered if you might consider... ' This is usually how it starts; the tentative preliminaries before the door creaks open and you are invited to enter an incestuous world of gala dinners, air kisses and warm champagne. And, once you have taken up a seat on the board of one of Britain's major arts or heritage institutions, there is no telling where it might lead, although, in all likelihood, it will simply lead straight to another seat on a board of governors or trustees.

The list of grandees, fixers and money-men at the top of our major cultural establishments is loaded with inter-relationships. There are husbands and wives, ex-husbands and ex-wives, children and business partners, along with a sober-minded smattering of refugees from scandal. And yet few of the thousands who regularly visit our national galleries, theatres and museums have any idea who sits on these boards, or what they do. For, while an executive head occasionally makes the news if debts mount up or the wrong kind of stone is used for a portico, the board has generally remained in the background.

But times are changing. Brash newcomers, who owe their seats to a growing public demand for representativeness and transparency, are beginning to take their places beside the old money around the oak tables at the RSC, the British Museum or the National Gallery.

This month, as Lord Hollick, the new chairman of the troubled South Bank Centre, reconstructs his board, the importance of getting the balance right is evident. Hollick, a former owner of Express Newspapers and the chief executive of United Business Media, must enlist some new blood and yet gain the help of the arts establishment if he wants to see plans for the hidebound arts complex come to fruition.

His first step has been to secure the services of Dame Vivien Duffield, doyenne of the fundraising dinner and former figurehead of the Royal Opera House. Duffield, whose partner is Sir Jocelyn Stevens, the former chairman of English Heritage, was not convinced she ought to take up the challenge. The bruises left by her departure from Covent Garden after a row with the chairman, Sir Colin Southgate, have not yet healed.

'I took quite a while to decide,' she says. 'In the end I said yes because I have always been fascinated by the possibilities of the site. A lot of members are appointed directly now. There is this mythical government list which people are supposed to put themselves on. I suspect it is still all done between the chairman and the board, though.'

The new South Bank board is stocked with similarly experienced hands. The vice-chair, Edward Walker-Arnott, a solicitor, also sits on the Royal National Theatre board, as does former civil servant Sir David Hancock, while Nigel Walmsley, former chairman of GMTV, also sits on the Ambassador Theatre Group board. Then there is the ubiquitous Alan Yentob, the BBC's director of drama and entertainment, who is also chairman of the Institute of Contemporary Arts and has just stepped down from the board of the Royal Court Theatre. The glamour quotient on the South Bank is provided by dancer and Arts Council member Deborah Bull and by the actress Joanna Lumley, who is also an official patron of the ICA.

But it is the financiers or 'grey men', as Duffield sees them, who really earn their seats on a board. 'It is difficult in this country,' she says. 'In the United States there is only one purpose for being there - money. Over here, we get such a large part of our money from government that it has a right to put people on a board, but very few practitioners will really add to a boardroom, with one or two notable exceptions. I know I am only there for the money. It is a case of what you can bring. It is "get, give or get off" in my view. You are not there to give artistic direction. The board is there to appoint the right chief executive and occasionally to either rein them in or help them expand.'

When it comes to pure influence, few can now do better than Hollick himself, whose wife, Sue, is the London representative on the new national Arts Council. While the Labour peer has declared that funding issues will not be discussed at home, it is hard to believe that such a close association will hurt. Lady Hollick, by the way, is also a member of Tate Modern's advisory council.

Along the river at the Royal National Theatre, the matrimonial connections are just as cosy, if not quite so conventional. Sir Christopher Hogg, the chairman of Reuters, is chairman of the board. He happens to be married to Dr Miriam Stoppard, whose first husband, the playwright Sir Tom Stoppard, sits on the RNT board.

Other notable RNT members include Greg Hutchings, who resigned as chief executive of Tomkins two years ago after an inquiry was launched into his spending on the company account; the author Ben Okri; Andre Ptaszynski of Andrew Lloyd Webber's Really Useful Company, and the broadcasters Sue McGregor and Joan Bakewell (she is also chairman of the British Film Institute and a member of the Film Council's board).

In fact, it seems that serious broadcasting is one of the surest ways for an arts industry outsider without a trace of financial acumen to win a place on a board. Reading the news, in particular, works a treat: Angela Rippon is chairman of English National Ballet, Jon Snow is on the boards of both the Tate and the National Gallery, while Sue Lawley sits on the board of the English National Opera.

The incoming artistic director of the National Theatre, Nicholas Hytner, has kept his place on the board at the Young Vic, while his mother, the irrepressible fundraiser Joyce Hytner, is on the board at its big sister, the Old Vic.

When it comes to glittering line-ups, it will always be hard to beat the Old Vic. Sally Greene, its chief executive, has even managed to lure Kevin Spacey into her boardroom, while Tina Brown, former editor of Vanity Fair, and John Malkovich are associate members. The only real rival to the Vic for star quality is the ICA, where Madonna is vice-president.

Whether names such as these have any real impact on the fortunes of an institution is less clear. For the playwright Bonnie Greer, a new appointee to the Royal Opera House board, it is arts practitioners who are crucial to putting together a successful board.

'You certainly need people with connections in the financial world, but we all have different strengths,' she said. 'It is old-fashioned to think artists are not experienced managers. We know an awful lot about how to make do. These are arts institutions, after all, and the first job is to produce and disseminate art. One can fund-raise and fund-raise, but if an institution is moribund, there is no point.'

Greer's chairman, Sir Colin Southgate, a former chairman of Thorn EMI, also sits on the National Gallery, while Lord Eatwell, also at the Opera House, is President of Queen's College, Cambridge and current frontrunner for the new job of chairing Ofcom. Gail Ronson, the wife of convicted fraudster Gerald Ronson, is on the board at Covent Garden too, as is Sir Peter Davis, chief executive of Sainsbury.

Among others who have managed to straddle at least two spheres of influence is Professor Dawn Ades, the art historian, who sits on the board of the Tate and the National Gallery. Her achievement, though, is as nothing compared to the influence of Sir Eddie Kulukundis, husband to the actress Susan Hampshire, who sits on the board of the Royal Shakespeare Company and the Hampstead Theatre, as well as being chairman of the Ambassador Theatre Group.

John Tusa, the former head of BBC's World Service, does just as well, serving on the board of trustees of the British Museum and on the board of the English National Opera. He is also managing director of the Barbican Arts Centre.

For beginners at this board game, publishing could be quite a good way in. Victoria Barnsley, chief executive of HarperCollins, sits on the board at the Tate, while Liz Calder, Bloomsbury's chief executive, chairs the board at the Royal Court Theatre.

A career in PR can do the trick too. 'It is jolly nice to have someone who has run a theatre,' says Duffield, 'because it is good to have someone who can tell you when you are being told lies, but you have to have accountants and marketing people.'

Architecture is also in vogue. Rick Mather, the man who drew up the ill-fated new masterplan for the South Bank Centre's redevelopment, is on the board at the Victoria & Albert Museum, while Sir Michael Hopkins sits on the British Museum's board of trustees. His wife and business partner Patricia is on the board of the National Gallery.

Finally, and perhaps inevitably, a political profile can do much to smooth your passage into the right circles. The former Culture Secretary Chris Smith sits on the boards of both the Donmar and National theatres; Labour peer Waheed Alli is on the English National Ballet board; Peter Mandelson is an associate board member of the Old Vic; and treasury advisor Ed Miliband, brother of the Education Minister David, is on the Royal Court's board.